Nuprl Lemma : scale-metric-leq-iff
∀[X:Type]. ∀[d1,d2:metric(X)]. ∀[c:{c:ℝ| r0 < c} ]. (c*d1 ≤ d2 ⇐⇒ d1 ≤ (r1/c)*d2)
Proof
Definitions occuring in Statement :
metric-leq: d1 ≤ d2,
scale-metric: c*d,
metric: metric(X),
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
metric-leq: d1 ≤ d2,
all: ∀x:A. B[x],
mdist: mdist(d;x;y),
scale-metric: c*d,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
prop: ℙ,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top
Latex:
\mforall{}[X:Type]. \mforall{}[d1,d2:metric(X)]. \mforall{}[c:\{c:\mBbbR{}| r0 < c\} ]. (c*d1 \mleq{} d2 \mLeftarrow{}{}\mRightarrow{} d1 \mleq{} (r1/c)*d2)
Date html generated:
2020_05_20-AM-11_39_14
Last ObjectModification:
2020_01_06-PM-00_22_39
Theory : reals
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